Bilinear Strichartz estimates on rescaled waveguide manifolds with applications

• Published in: arXiv.org 2024-12
• Main Authors: Chen, Qionglei, Song, Yilin, Yang, Kailong, Zhang, Ruixiao, Zheng, Jiqiang
• Format: Article
• Language: English
• Subjects: Decoupling; Estimates; Nonlinearity; Waveguides; Well posed problems
• ISSN: 2331-8422

We focus on the bilinear Strichartz estimates for free solutions to the Schr\"odinger equation on rescaled waveguide manifolds \(\mathbb{R} \times \mathbb{T}_\lambda^n\), \(\mathbb{T}_\lambda^n=(\lambda\mathbb{T})^n\) with \(n\geq 1\) and their applications. First, we utilize a decoupling-type estimate originally from Fan-Staffilani-Wang-Wilson [Anal. PDE 11 (2018)] to establish a global-in-time bilinear Strichartz estimate with a `\(N_2^\epsilon\)' loss on \(\mathbb{R} \times \mathbb{T}^n_\lambda\) when \(n\geq1\), which generalize the local-in time estimate in Zhao-Zheng [SIAM J. Math. Anal. (2021)] and fills a gap left by the unresolved case in Deng et al. [J. Func. Anal. 287 (2024)]. Second, we prove the local-in-time angularly refined bilinear Strichartz estimates on the 2d rescaled waveguide \(\mathbb{R} \times \mathbb{T}_\lambda\). As applications, we show the local well-posedness and small data scattering for nonlinear Schr\"odinger equations with algebraic nonlinearities in the critical space on \(\mathbb R^m\times\mathbb{T}^n\) and the global well-posedness for cubic NLS on \(\mathbb{R} \times \mathbb{T}\) in the lower regularity space \(H^s\) with \(s>\frac{1}{2}\).